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This paper is devoted to the location of public facilities in a metric space. Selfish agents are located in this metric space, and their aim is to minimize their own cost, which is the distance from their location to the nearest facility. A central authority has to locate the facilities in the space, but she is ignorant of the true locations of the agents. The agents will therefore report their locations, but they may lie if they have an incentive to do it. We consider two social costs in this paper: the sum of the distances of the agents to their nearest facility, or the maximal distance of an agent to her nearest facility. We are interested in designing strategy-proof mechanisms that have a small approximation ratio for the considered social cost. A mechanism is strategy-proof if no agent has an incentive to report false information. In this paper, we design strategyproof mechanisms to locate n − 1 facilities for n agents. We study this problem in the general metric and in the tree metric spaces. We provide lower and upper bounds on the approximation ratio of deterministic and randomized strategy-proof mechanisms.

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This paper deals with problems which fall into the domain of selfish scheduling: a protocol is in charge of building a schedule for a set of tasks without directly knowing their length. The protocol gets these informations from agents who control the tasks. The aim of each agent is to minimize the completion time of her task while the protocol tries to minimize the maximal completion time. When an agent reports the length of her task, she is aware of what the others bid and also of the protocol's algorithm. Then, an agent can bid a false value in order to optimize her individual objective function. With erroneous information, even the most efficient algorithm may produce unreasonable solutions. An algorithm is truthful if it prevents the selfish agents from lying about the length of their task. The central question in this paper is: "How efficient a truthful algorithm can be? We study the problem of scheduling selfish tasks on parallel identical machines. This question has been raised by Christodoulou et al [8] in a distributed system, but it is also relevant in centrally controlled systems. Without considering side payments, our goal is to give a picture of the performance under the condition of truthfulness.

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The distributed nature of the grid results in the problem of scheduling parallel jobs produced by several independent organizations that have partial control over the system. We consider systems in which each organization owns a cluster of processors. Each organization wants its tasks to be completed as soon as possible. In this paper, we model an off-line system consisting of N identical clusters of m processors. We show that it is always possible to produce a collaborative solution that respects participants' selfish goals, at the same time improving the global performance of the system. We propose an algorithm (called MOLBA) with a guaranteed worst-case performance ratio on the global makespan, equal to 4. Next, we show that a better bound (equal to 3) can be obtained in a specific case when the last completed job requires at most m/2 processors. Then, we derive another algorithm (called ILBA) that in practice improves the proposed, guaranteed solution by further balancing the schedules. Finally, by an extensive evaluation by simulation, we show that the algorithms are efficient on typical instances. } we denote the set of independent organizations forming the grid. Each organization O k owns a cluster M k . By M we denote the set of all clusters. Each cluster M k has m k Proposition 4. ILBA does not delay any job when compared with the base schedule.Proof. As ILBA considers clusters sequentially, it is sufficient to show that the proposition holds for any cluster M k (as the jobs from the following clusters do not modify the jobs already scheduled).Let us assume that jobs are numbered according to non-decreasing start times in the base schedule.The proof is by induction on the scheduled jobs. In ILBA, a scheduled job J i cannot be influenced by jobs J i+1 , . . . , J n scheduled afterward (or by jobs incoming from different clusters in the subsequent parts of the algorithm). Note also that at this phase of the algorithm, no new job is allocated to M k . Thus, it is sufficient to show that if jobs J 1 , . . . , J i−1 are completed no later than in the original schedule, J i is also not delayed.The proposition trivially holds for J 1 , as the first job will be started at t = 0. COOPERATION IN MULTI-ORGANIZATION SCHEDULING 915Let us now assume that jobs J 1 , . . . , J i−1 are scheduled no later than their start times in the original schedule. Consequently, J 1 , . . . , J i−1 finish no later than in the original schedule. Thus, at J i 's original start time, the number of free processors is at least the same as in the original schedule. Hence, J i can be scheduled at its original start time. J i is migrated only if it can be started earlier than this original start time. Consequently, J i finishes at the latest when it finished in the base schedule. SIMULATIONSIn this section, we carry out the experimental evaluation of the proposed algorithms. We start with performance measures and the methods used to generate workload. Then, we show the performance of the considered algorithms. MethodsWe compare three algorit...

International audienceWe consider the problem of designing truthful mechanisms for scheduling \em selfish tasks (or agents)-whose objective is the minimization of their completion times- on parallel identical machines in order to minimize the \em makespan. A truthful mechanism can be easily obtained in this context (if we, of course, assume that the tasks cannot shrink their lengths) by scheduling the tasks following the increasing order of their lengths. The quality of a mechanism is measured by its approximation factor (price of anarchy, in a distributed system) w.r.t. the social optimum. The previous mechanism, known as SPT, produces a $(2-1/m)$-approximate schedule, where $m$ is the number of machines. The central question in this paper is the following: \em ``Are there other truthful mechanisms with better approximation guarantee (price of anarchy) for the considered scheduling problem?'' This question has been raised by Christodoulou et al \citekoutsoupias in the context of coordination mechanisms, but it is also relevant in centrally controlled systems. We present (randomized) truthful mechanisms for both the centralized and the distributed settings that improve the (expected) approximation guarantee (price of anarchy) of the SPT mechanism. Our centralized mechanism holds for any number of machines and arbitrary schedule lengths, while the coordination mechanism holds only for two machines and schedule lengths that are powers of a certain constant

We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations, where m is constant. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan C max , which is the maximum completion time.An extended abstract of this work has been accepted at 392 Algorithmica (2010) 58: [391][392][393][394][395][396][397][398][399][400][401][402][403][404] The general case of the problem is inapproximable unless P = NP; hence, we study a suitable strongly NP-hard restriction, namely the case where at least one machine is always available. For this setting we contribute approximation schemes, complemented by inapproximability results. The approach is based on algorithms for multiple subset sum problems; our technique yields a PTAS which is best possible in the sense that an FPTAS is ruled out unless P = NP. The PTAS presented here is the first one for the problem under consideration; so far, not even for well-known special cases approximation schemes have been proposed. Furthermore we derive a low cost algorithm with a constant approximation ratio and discuss FPTASes for special cases as well as the complexity of the problem if m is part of the input.